Theory of Fuzzy Sets
"Fuzzy sets arise when a set partially contains an element, ..."
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To explain what a fuzzy set is, let me quote from Bart Kosko. In Fuzzy Thinking: The New Science of Fuzzy Logic, he writes:
... Fuzzy sets arise when a set partially contains an element, as when an audience [a group of people in a room, say] contains a somewhat happily employed person or when a barrel contains a somewhat rotten apple or when a chromosome contains a somewhat mutated gene. In some sense the set is not fuzzy but its elements are fuzzy. The elements all have some property to some degree. I call this elementhood. The old fuzzy or multivalued logic is all about elementhood.
What happens when one set contains another set? Put a little box in a big box. Then the big box contains the little box. It contains the little box 100%. Can that containment take on degrees? Sure it can. Put the little box only half way in the big box. Then the big box contains the little box only 50%. The little box is both in and outside the big box.
I called this fuzzy containment subsethood, the degree to which one set is a subset of another set. Traditional fuzzy theory assumed subsethood was bivalent, all or none, 100% or 0%. That seemed as extreme as any other black-white claim. Very tall men made up a 100% subset of tall men. That I could buy. Every very tall man is tall. But the old view said that tall men made up only a 0% subset of very tall men. That I could not buy. It was a matter of degree. Every tall man is very tall to some degree, often to a very small degree.
According to the dictionary, to contain is to hold or accommodate. That's the key. All three terms -- contain, hold, accommodate -- mean to have within or to have the capacity for having within. But there is nothing explicit in the meaning that says that holding within has to be totally within. The holding could be only partial and still satisfy the definition. So you can readily accept the idea of one set being only partially in another set. The range from none to all would then be from 0% to 100%. Therefore the degree of containment would range from 0 for 0% to 1 for 100%).
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In ordinary set theory a set is defined in terms of its members. We say that a set, A, is the set of elements, x -- given the proviso that x satisfies some function, f(x). That is:
A = {x | f(x)}
Or A is the set of elements, x, such that f(x).
Finite Sets
To deal with fuzzy sets, the idea of partial containment has to be considered. An example is given by Kurt J Schmucker (in Fuzzy Sets, Natural Language, Computations, and Risk Analysis). In his example the universe is the set {a, b, c, d, e, f}, and one possible fuzzy subset, A, could be defined in the following way:
a is present with degree of membership 1.0
b is present with degree of membership .9
c is present with degree of membership .2
d is present with degree of membership .8
e is present with degree of membership 1.0
f is present with degree of membership 0
This longwinded expression can be written equivalently as:
A = {1/a, .9/b, .2/c, .8/d, 1/e},
where each element is juxtaposed next to its degree of containment and elements with 0 degree containment are omitted.
Infinite Sets
The same general idea applies to infinite sets. But now the elements and degree of membership can't be listed, so they have to be defined using functions. This can be written as the set:
Y = {m(x)/x | f(x)}
In this definition, m(x) is the function identifying the degree of membership of each x in Y. And f(x) is the function that defines the universe of elements making up the membership of x.
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As with ordinary sets, the operations involving fuzzy sets include the union, intersection, and complement, but now it's less clear how the operations are to be defined.
Set Inclusion
First consider the general problem of set inclusion. What does it mean for a fuzzy set to be included in another fuzzy set?
Sets are defined in terms of their members, as we see here, and subsets are collections of the members. This is the case whether we're dealing with ordinary sets or fuzzy sets, but with fuzzy sets we have to take the degree of membership into account. Say the set, A, is contained in set, B. If an element, a, is a member of A, to degree m(a), is it a member of B to the same degree? Not necessarily.
Consider the set of rose-colored apples, A, and the set of red apples, B. In ordinary sets, rose colored apples would form a subset of red apples and we would say that the former is included in the latter. All rose apples are red apples. A Ì B. If x is a member of A, x is also a member of B.
In fuzzy sets, though, a set, A, may be only partially included in another set, B. So a member with a certain degree of membership in A would be likely to have a different degree of membership in B. A light rose apple might be in A to a medium degree, but in B only to a very slight degree. Light rose apples are OK in A, but they barely qualify for membership in B.
Union of Fuzzy Sets
Operations for sets are ways to combine sets, or ways to generate new sets out of old ones. Operations on sets correspond to the operations you find in arithmetic, which you know as addition, subtraction, multiplication, and division. There are also operations in logic that do similar kinds of things. The difference is that set operations deal with sets, and arithmetic operations work on numbers, and logic operations have to do with combining true or false statements. In set theory, then, you find the operators, È (read as union, join, or sum), Ç (read as intersection, meet, or product), and Comp (or complement).
In ordinary sets the union of sets A and B is the set of all members that are either in A or in B and is written:
A È B
So, if x is a member of A, then x is a member of A È B, and if y is a member of B, then y is a member of A È B.
With fuzzy sets, though, x and y would each have their own degree of membership in their respective sets, so their degree of membership in the union isn't predetermined. If x is in A to degree .7, it needn't be in A È B to the same degree. Indeed, A itself could be in A È B only to some degree. Similarly with B in A È B.
Following Schmucker in a formulation proposed by L A Zadeh, the union is expressed as:
A È B = {max(a(x), b(x))/x | x is an element of U}
In this formulation a(x) is the degree of membership of x in A, and b(x) is the degree of membership of x in B. To get the membership of x in A È B, you have to compare the membership of x in each of the two sets, A and B, and take the larger of the two values as the degree of membership of x in A È B.
Intersection of Fuzzy Sets
Similarly, the intersection of A and B in ordinary sets is the set of all members of A that are also members of B and is written as:
A Ç B
So, if x is a member of A and if x is a member of B, then x is a member of A Ç B.
In fuzzy sets, however, x may not have the same degree of membership in the two sets taken separately, so what might be the degree of membership in their intersection? The answer isn't obvious. But here's how Zadeh proposed to define it:
A Ç B = {min(a(x), b(x))/ | x is an element of U}
Since the degree of membership of x in A could be different than the degree of membership of x in B, you have to compare them and use the smaller of the two values to define the degree of membership of x in the intersection.
Complement of a Fuzzy Set
The complement of a set is an operation that's performed only on one set -- not two, as in the union and intersection. That unary operator is known as complementation, Comp(A). Given a set, A, you take its complement relative to the universe of discourse, U. So, if A consists of the set {x}, with x drawn from U, then the complement of A is:
Comp(A) = {x | x Ï A}
which consists of all the members of U that are not in A.
That's okay for ordinary sets, but what about fuzzy sets? If x is a member of A to some degree a(x), then it has to be in the complement of A for the remaining degree. That is, x would be in Comp (A) to degree (1 - a(x)). So the complement of A would be the set:
Comp(A) = {(1 - a(x))/x | x is in A}
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The characteristic function of fuzzy sets generalizes on the characteristic function of ordinary sets. It is a mapping from the whole of the universe, U, but it differs from the function for ordinary sets in that the mapping goes to a portion of the real line from 0 to 1 instead of just to the points 0 and 1. For a fuzzy subset A, the line portion is expressed using solid brackets:
CA(x): U ® [0, 1]
So the characteristic function is the degree of membership function, m(x).
Recall that 1 and 0 were associated with the truth values true and false. Since any segment of the real line has an infinite number of points, the mapping of the fuzzy characteristic function goes to an infinite number of points, each of which defines a degree of membership in the set.
A similar association with truth values therefore gives us an infinite number of them, so we now have an infinite number of possible truth values, which can range anywhere from 1 to 0.
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